$\Sigma$-algebraically compact modules and $L_{\omega _1\omega}$-compact cardinals

Abstract

We prove that the property $\mathrm{Add}(M)\subseteq \mathrm{Prod}$ characterizes $\Sigma$-algebraically compact modules if $|M|$ is not $\omega$-measurable. Moreover, under a large cardinal assumption, we show that over any ring $R$ where $|R|$ is not $\omega$-measurable, any free module $M$ of $\omega$-measurable rank satisfies $\mathrm{Add}\subseteq \mathrm{Prod}$, hence the assumption on $|M|$ cannot be dropped in general (e.g. over small non-right perfect rings).

In this way, we extend results from a recent paper by Simion Breaz.

Keywords

• ($\Sigma$)-algebraically compact module
• strongly compact cardinal
• large ultrapower